curve #4264

y² + xy + y = x³ − 836303671819227360448892158241356598928678x + 294386348669829639707635403888781174306416078465529045951121948

a-invariants: [1, 0, 1, -836303671819227360448892158241356598928678, 294386348669829639707635403888781174306416078465529045951121948]
c-invariants: [40142576247322913301546823595585116748576521, -254349805250732808707457202824277918970695812029610473376892228069]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 310185621680651834905251713151662581225990024407184633137813567866714832202772684700109205190
discriminant (Δ): -4059714867846464612952509579831371227530801741091951999101234159870332617575772149629696350028177466955544688612536096312500
Faltings height: 22.7771
naive height: 301.2031
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 41, 53, 59, 83, 89, 2011161233376865227863209973974465563186794920446510866861224845742039487039
regulator: 18421311845773731515083879.98028731576732728664879916606541034291584544626532793492962783
submitted by: apkorte
submitted at: 2026-10-15 21:54:58 UTC
last updated: 2026-10-15 21:54:58 UTC

Witness: 25 independent points log in to add more points →

(511379322488021094824, 669434398307403899721512850075)
(726129004915446929084, 8365595858752351292629657912335)
(55120435932686342738924/25, -11969095558138296059029440934962492/125)
(523265536976173710344, 226125185414371531248786233595)
(-851170966621133202601, 19737203074288440094091088521175)
(545905486034012307647, -728291485944177624427091371233)
(2003358660892190131031/4, -8627786617422603193976740937115/8)
(245315004920047561124, 10197618926750918747107893657975)
(773181364976835731399, 10487470616623709676103529917425)
(1066303509719447696471/4, -76059582852469742553886142162475/8)
(80057016241196572360411/9, 22536348190332123285757767056169305/27)
(776086027071977520524, -10620111022669795638854486510700)
(-1055760583765885752976, -734812114048395815203503695700)
(508714839085709323784, 772641350245262457622466743035)
(571444960249395515909, 1757784477120389421596258714535)
(509945675928699735760559/961, 4909043627806196370461861007842385/29791)
(523920217465390457099, -205090129708185143328605281725)
(987414742112950399874, 20768369539665126985790734763400)
(6375603367935464596949, 504103164963805638291255047348475)
(425303614448117053274, 3953939542528384495515327511800)
(62969704858987737146279/121, -434139571386113181806238383153325/1331)
(61900725605932030435379/121, 880882533906642422807078888542275/1331)
(542365797497305357019, -588660158689383809729830845165)
(536586432781071528299, 365777612229931012882180935075)
(2309063290191667923539, 103318489093150477131199905849345)

Commentary

Specialization at T = -4927/4396 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -234114040453021356506112T^8 - 272002889947768518314880T^7 + 686028086039421713758320T^6 + 1142195364874563315587280T^5 + 627637229082985727538360T^4 - 984449918457333810927360T^3 - 2633542817168016219118320T^2 - 1006973197204579283167560T - 114715927313180034201675, A6 = 48308501179046578125484559991177216T^12 + 57069826702680132557515086694133760T^11 - 311131671782769127044180046463351040T^10 - 240521895686500160053739383108517760T^9 + 1147866319247799475598436207881897040T^8 + 683399028950475828317237644205746752T^7 - 2372093631572614744214600133762992400T^6 - 1765634230326260432804074074782197200T^5 + 1564909970672513896400523764894111640T^4 + 1894740665595018649008096627125850240T^3 + 882898465151004944410908107862516576T^2 + 191456284444900580427321522994041240T + 15250200281398537130338745700198150. It is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-15 21:54:58 UTC · history